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SPECULOOS-South, an observatory composed of four independent 1m robotic telescopes, located at ESO Paranal, Chile, started scientific operation in January 2019. This Southern Hemisphere facility operates as part of SPECULOOS, an international network of 1m-class telescopes surveying for transiting terrestrial planets around the nearest and brightest ultra-cool dwarfs. To automatically and efficiently process the observations of SPECULOOS-South, and to deal with the specialised photometric requirements of ultra-cool dwarf targets, we present our automatic pipeline. This pipeline includes an algorithm for automated differential photometry and an extensive correction technique for the effects of telluric water vapour, using ground measurements of the precipitable water vapour. Observing very red targets in the near-infrared can result in photometric systematics in the differential lightcurves, related to the temporally-varying, wavelength-dependent opacity of the Earth's atmosphere. These systematics are sufficient to affect the daily quality of the lightcurves, the longer time-scale variability study of our targets and even mimic transit-like signals. Here we present the implementation and impact of our water vapour correction method. Using the 179 nights and 98 targets observed in the I+z' filter by SPECULOOS-South since January 2019, we show the impressive photometric performance of the facility (with a median precision of ~1.5 mmag for 30-min binning of the raw, non-detrended lightcurves) and assess its detection potential. We compare simultaneous observations with SPECULOOS-South and TESS, to show that we readily achieve high-precision, space-level photometry for bright, ultra-cool dwarfs, highlighting SPECULOOS-South as the first facility of its kind.